NONPOSITIVE EIGENVALUES OF HOLLOW, SYMMETRIC, NONNEGATIVE MATRICES

Among hollow, symmetric n-by-n nonnegative matrices, it is shown that any number k, 2 <= k <= n - 1 of nonpositive eigenvalues is possible. However, as n grows, small numbers of nonpositive eigenvalues become increasingly rare. In particular, if there is only a given finite number of distinct off-diagonal entries, the minimum number of nonpositive eigenvalues (among n-by-n hollow, symmetric, nonnegative matrices) grows with n, and this remains so if just the ratio of the smallest positive off-diagonal entry to the largest is bounded away from zero. Nonetheless, every n-by-n hollow, symmetric, nonnegative matrix with positive off-diagonal entries and with two nonpositive eigenvalues may be embedded in an (n + 1)-by-(n + 1) one as a principal submatrix. This statement does not hold without the requirement of positive off-diagonal entries. Our proof recognizes some special, anti-Morishima structure of Schur complements which may be of independent interest. Relations to the independence number of a graph are also presented.